Solving for the interior angles of a triangle

Every triangle has three interior angles

The interior angles of a triangle are the three angles on the inside of a triangle. These three angles always sum to $180{}^\circ$ .

$x{}^\circ +y{}^\circ +z{}^\circ =180{}^\circ$

There are a few other angle relationships we need to remember:

The angle measures that form a straight line add to $180^\circ$ , so $z^\circ+w^\circ=180^\circ$ .

Vertical angles are congruent, so $x^\circ=y^\circ$ .

Let’s start by working through an example.

Example

What is the value of $x$ ?

We know that these two angles lie on a straight line:

We can find the measure of the angle by subtracting $180{}^\circ -142{}^\circ =38{}^\circ$ .

The three angles inside a triangle sum to $180{}^\circ$ , so

$180{}^\circ -104{}^\circ -38{}^\circ =38{}^\circ$

Now we can see that $x^\circ$ and $38^\circ$ lie on a straight line, so

$x{}^\circ +38{}^\circ =180{}^\circ$

$x{}^\circ =142{}^\circ$

Let’s try another one.

Example

What is the value of $y$ ?

The three interior angles of a triangle sum to $180{}^\circ$ , so

$m\angle 1+19{}^\circ +41{}^\circ =180{}^\circ$

$m\angle 1=120{}^\circ$

Vertical angles are congruent, so

$m\angle 1=m\angle 2=120{}^\circ$

Again, the three angles of a triangle sum to $180{}^\circ$ , so we can say

$m\angle 2+38^\circ +y^\circ =180^\circ$

$120^\circ +38^\circ +y^\circ =180^\circ$

$y^\circ =22^\circ$